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Simetrical comments on The Aumann's agreement theorem game (guess 2/3 of the average) - Less Wrong
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Comments (149)
If everybody reasons as you describe then everyone will guess 1/∞ and everyone will tie. You can't get closer to 2/3 of an infinitesimal than an infinitesimal, so it's stable.
Disclaimer: I'm not mathy. Maybe you actually can get closer to 2/3 of an infinitesimal than an infinitesimal.
The question required us to provide real numbers, and infinitesimals are not real numbers. Even if you allowed infinitesimals, though, 0 would still be the Nash equilibrium. After all, if 1/∞ is a valid guess, so is (1/∞)*(2/3), etc., so the exact same logic applies: any number larger than 0 is too large. The only value where everyone could know everyone else's choice and still not want to change is 0.